Quantum wells have been developed in the context of solar cells mainly to tailor the absorption edge of the sub-cells in multi-junction devices to their optimum values (1). Typically, achieving the proper performance requires a delicate trade-off between carrier collection and light absorption (2,3). Solcore includes a simplified QW structure in the PDD solver in order to calculate the performance of solar cells containing them. Contrary to other programs like Nextnano, Solcore does not solve the Schrödinger equation and the PDD equations self-consistently: first, the energy levels of the quantum wells are solved using a flat-band condition, considering also the strain in the materials, and then an effective band structure is used to solve the transport equations in a bulk-like fashion. This is illustrated in the next figure:
From the perspective of the PDD solver, the actual bandgap and electron affinity of each layer in a quantum well depend on the energy levels, i.e. the minimum energy for electrons is not the band edge of the conduction band, but the ground confined level. The same applies to holes, with the actual band edge being the maximum between the ground states of light holes and heavy holes. The resulting band profiles used in the PDD solver are shown in the right end of the figure.
To use QWs in the PDD solver, we create an effective electron affinity and bandgaps for all layers in the QW. For the barriers, the electron affinity and band gap are the same as they are in bulk, modified by the strain, if necessary. For interlayers, if present, it depends on what is higher, the band edges of the interlayer or the confined carrier levels.
The density of states and the absorption profile need to be modified in a similar way. For the density of states:
- Barriers have the bulk density of states and absorption profile.
- Interlayers only have the bulk density of states above the barrier and the bulk absorption from the barrier energy and zero below that.
- Wells have all the density of states associated with the confined states and the bulk density of states above the barrier, while they have the absorption of the confined levels below the barrier energy and of the bulk above it.
These simplifications are similar to those in Nelson et al. (4) and in Cabrera et al. (5) and allow us to keep the bulk-like form of the carrier densities in the drift diffusion equations under the Boltzmann approximation. A more rigorous treatment will be necessary in the presence of tunnel transport across a supperlattice, tunnel escape from the QWs to the barriers - possible in the presence of high electric fields - and in the case of very deep QWs, when carrier escape from the less confined levels might be possible but not from the deeper ones. In these situations, a set of rate equations linking the different levels, as well as a self-consistent solution of the transport and Schrödinger equations would be required, besides using more advanced methods such as a non-equilibrium Green’s functions (NEGF) formalism (6).
This module defines a class derived from solcore.Structure that allows to solve the Schrodinger equation and the kp model. It also prepares the properties of the structure (bandedges, efective density of states (DOS), etc) in order to have a meaningful set of properties for the PDD. Without this preparation, the structure is just a collection of layers with bulk-like properties, as it is illustrated in the figure:
In this case, "effective_QW" is a list of 200 layers (5 layers per QW unit repeated 40 times) but rather than the bulk material properties, they include effective properties as a result of the quantum calculation.
solcore.poisson_drift_diffusion.QWunit
Thomas, T., Wilson, T., Führer, M., Alonso-Álvarez, D., Ekins- Daukes, N.J., Lackner, D., Kailuweit, P., Philipps, S.P., Bett, A.W., Toprasertpong, K., Sugiyama, M., Okada, Y.: Potential for reaching 50% power conversion efficiency using quantum heterostructures. In: 6th World Conference on Photovoltaic Energy Conversion, pp. 1–2 (2014)↩
Alonso-Álvarez, D., Führer, M., Thomas, T., Ekins-Daukes, N.: Elements of modelling and design of multi-quantum well solar cells. In: 2014 IEEE 40th Photovoltaic Specialists Conference (PVSC), pp. 2865–2870 (2014)↩
Alonso-Álvarez, D., Ekins-Daukes, N.J.: Quantum wells for high- efficiency photovoltaics. In: Freundlich, A., Lombez, L., Sugiyama, M. (eds.) Physics, Simulation, and Photonic Engineering of Pho- tovoltaic Devices V, vol. 9743, p. 974311. SPIE OPTO, San Francisco, CA (2016). https://doi.org/10.1117/12.2217590↩
- Nelson, M. Paxman, K. W. J. Barnham, J. S. Roberts, and C. Button, “Steady-state carrier escape from single quantum wells,” IEEE J. Quantum Electron., vol. 29, no. 6, pp. 1460–1468, 1993.
- Cabrera, J. C. Rimada, J. P. Connolly, and L. Hernandez, “Modelling of GaAsP/InGaAs/GaAs strain-balanced multiple-quantum well solar cells,” J. Appl. Phys., 113, 024512, (2013).
Aeberhard, U.: Quantum-kinetic perspective on photovoltaic device operation in nanostructure-based solar cells. J. Mater. Res. 33, 373–386 (2018)↩